With a history starting at the time of Newton, the numerical solution of the one-dimensional Riemann integrals is the oldest, the most studied, and as yet unsolved problem of the numerical analysis. A number of pitfalls like the Runge and the Gibbs phenomena illustrate the difficulties which had to be overcome during the classical period of this topic.

The advent of the modern digital computers raised great hopes that the high computing speed associated with smart computing schemes like the automatic adaptive quadrature will get rid of the stumbling blocks against the derivation of reliable numerical outputs under the only condition that the integral exists and is finite. The dream that computer experiments would be possible for the simulation of physical processes involving sudden changes of the integrand properties under the variation of the system parameters remained only a dream.

Less than a decade ago, a pitfall was pinpointed in the floating point computation of the integrals, namely, that the algebraic degree of precision ceases to be an invariant of a given interpolatory quadrature sum [1]. Straightforward consequences of this discovery were the need of a multiscale approach to the Bayesian solution of the integrals with respect to the length of the integration domain [2] and to the range of variation of the integrand function [3].

In the present lecture we discuss the significant progress of the multiscale approach coming from the concomitant use of two kinds of quadrature rules: (1) an m-panel quadrature rule of high algebraic degree of precision (the best candidate being a Clenshaw-Curtis quadrature rule spanned at the extremal points of a Chebyshev polynomial of the first kind and of the polynomial degree 32, hence m = 33 (CC-32)) [3] and (2) a local generalized Simpson rule at the predefined abscissas of the CC-32 rule.

The basic ingredients of the Simpson rule (local integrand values, slopes and curvatures) enable fast solution of exceptional cases, quick localization of the difficult spots of the integrand function and rid of the Gibbs phenomenon. The global CC-32 rule gets quick solution of regular integrand for which good interpolatory polynomial approximations of large polynomial degree hold.

References
[1] S. Adam, Gh. Adam, LNCS 7125, Springer, (2012) pp. 189-194.
[2] Gh. Adam, S. Adam, EPJ Web of Conferences, 108, 01001 (2016) pp.1-8.
[3] Gh. Adam, S. Adam, EPJ Web of Conferences, 173, 01001 (2018) pp.1-8.